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Mirrors > Home > ILE Home > Th. List > rexlimdv3a | GIF version |
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2551. (Contributed by NM, 7-Jun-2015.) |
Ref | Expression |
---|---|
rexlimdv3a.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
rexlimdv3a | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimdv3a.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) | |
2 | 1 | 3exp 1181 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
3 | 2 | rexlimdv 2551 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 ∈ wcel 1481 ∃wrex 2418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-nf 1438 df-ral 2422 df-rex 2423 |
This theorem is referenced by: resqrtcl 10833 |
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